On the hypersurfaces of constant curvature in S^{n+1} with boundary

We prove the existence of a hypersurface of constant Gauss curvature K in 𝕊ⁿ⁺¹ with Γ as a boundary under the condition that Γ bounds a certain locally convex hypersurface where K is a given positive constant and Γ is a disjoint collection Γ = {Γ₁, …, Γ_m} of closed smooth embedded (n-1) dimensional submanifolds of 𝕊ⁿ⁺¹. We prove some important local properties of locally convex hypersurfaces then use this and a Perron method to show the convergence of an area minimizing sequence of hypersurfaces. Regularity of resulting hypersurface is studied.

We are also interested in the extension of the above result to hypersurfaces satisfying more general curvature condition and we need first the existence theorem to the Dirichlet problem of some fully nonlinear elliptic equation. To apply the Evans-Krylov theory and standard existence arguments, we establish a priori estimates for principal curvatures of the surface which is the graph of the solution to the above mentioned PDE.